+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground/notation/relations/rat_3.ma".
-include "ground/arith/nat_plus.ma".
-include "ground/arith/nat_lt.ma".
-include "ground/relocation/mr2.ma".
-
-(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
-
-inductive at: mr2 → relation nat ≝
-| at_nil: ∀i. at (◊) i i
-| at_lt : ∀cs,l,m,i1,i2. i1 < l →
- at cs i1 i2 → at (❨l, m❩;cs) i1 i2
-| at_ge : ∀cs,l,m,i1,i2. l ≤ i1 →
- at cs (i1 + m) i2 → at (❨l, m❩;cs) i1 i2
-.
-
-interpretation "application (multiple relocation with pairs)"
- 'RAt i1 cs i2 = (at cs i1 i2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact at_inv_nil_aux: ∀cs,i1,i2. @❪i1, cs❫ ≘ i2 → cs = ◊ → i1 = i2.
-#cs #i1 #i2 * -cs -i1 -i2
-[ //
-| #cs #l #m #i1 #i2 #_ #_ #H destruct
-| #cs #l #m #i1 #i2 #_ #_ #H destruct
-]
-qed-.
-
-lemma at_inv_nil: ∀i1,i2. @❪i1, ◊❫ ≘ i2 → i1 = i2.
-/2 width=3 by at_inv_nil_aux/ qed-.
-
-fact at_inv_cons_aux: ∀cs,i1,i2. @❪i1, cs❫ ≘ i2 →
- ∀l,m,cs0. cs = ❨l, m❩;cs0 →
- i1 < l ∧ @❪i1, cs0❫ ≘ i2 ∨
- l ≤ i1 ∧ @❪i1 + m, cs0❫ ≘ i2.
-#cs #i1 #i2 * -cs -i1 -i2
-[ #i #l #m #cs #H destruct
-| #cs1 #l1 #m1 #i1 #i2 #Hil1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_introl, conj/
-| #cs1 #l1 #m1 #i1 #i2 #Hli1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_intror, conj/
-]
-qed-.
-
-lemma at_inv_cons: ∀cs,l,m,i1,i2. @❪i1, ❨l, m❩;cs❫ ≘ i2 →
- i1 < l ∧ @❪i1, cs❫ ≘ i2 ∨
- l ≤ i1 ∧ @❪i1 + m, cs❫ ≘ i2.
-/2 width=3 by at_inv_cons_aux/ qed-.
-
-lemma at_inv_cons_lt: ∀cs,l,m,i1,i2. @❪i1, ❨l, m❩;cs❫ ≘ i2 →
- i1 < l → @❪i1, cs❫ ≘ i2.
-#cs #l #m #i1 #m2 #H
-elim (at_inv_cons … H) -H * // #Hli1 #_ #Hi1l
-elim (nlt_ge_false … Hi1l Hli1)
-qed-.
-
-lemma at_inv_cons_ge: ∀cs,l,m,i1,i2. @❪i1, ❨l, m❩;cs❫ ≘ i2 →
- l ≤ i1 → @❪i1 + m, cs❫ ≘ i2.
-#cs #l #m #i1 #m2 #H
-elim (at_inv_cons … H) -H * // #Hi1l #_ #Hli1
-elim (nlt_ge_false … Hi1l Hli1)
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem at_mono: ∀cs,i,i1. @❪i, cs❫ ≘ i1 → ∀i2. @❪i, cs❫ ≘ i2 → i1 = i2.
-#cs #i #i1 #H elim H -cs -i -i1
-[ #i #x #H <(at_inv_nil … H) -x //
-| #cs #l #m #i #i1 #Hil #_ #IHi1 #x #H
- lapply (at_inv_cons_lt … H Hil) -H -Hil /2 width=1 by/
-| #cs #l #m #i #i1 #Hli #_ #IHi1 #x #H
- lapply (at_inv_cons_ge … H Hli) -H -Hli /2 width=1 by/
-]
-qed-.